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Statement

Notation (pp. 508, 510): $\sigma(n)$ is the sum of the divisors of $n$; $n$ is abundant if $\sigma(n)>2n$, semiperfect if it is a sum of distinct proper divisors of $n$, weird if it is abundant and not semiperfect, and primitive weird if it is weird and not a multiple of another weird number. The abundance is $\Delta(n)=\sigma(n)-2n$.

Theorem 1 (p. 509). Let $k$ be a positive integer and let $a$ and $b$ be positive odd integers such that $p=2^{k+2}-a$ and $q=2^{k+2}+b$ are both prime. If

$$b+3<a<2^{(k-1)/2},$$

then $n=2^kpq$ is a primitive weird number.

The hypotheses force $k\ge6$: $a$ is odd and exceeds $b+3\ge4$, so $a\ge5$, and $5<2^{(k-1)/2}$ needs $k\ge6$. The paper notes (p. 509) that the least integer the theorem yields comes from $(a,b,k)=(5,1,6)$, namely $2^6(2^8-5)(2^8+1)=4\,128\,448$, the 32nd primitive weird number, and that no other triple has $k\le7$.

Conditional consequence (p. 509, unnumbered). Let $p_n$ be the $n$th prime. The paper states that if $p_{n+1}-p_n<0.1\,p_n^{1/2}$ for all sufficiently large $n$, then Theorem 1 gives infinitely many primitive weird numbers of the form $2^kpq$. It adds that Cramér's conjecture $p_{n+1}-p_n=O((\log p_n)^2)$, or the much weaker conjecture it attributes to Gonek, that for every $\varepsilon>0$ one has $p_{n+1}-p_n<p_n^{\varepsilon}$ for all large $n$, suffices, and (p. 510) that the Baker--Harman--Pintz bound $p_{n+1}-p_n<p_n^{0.525}$ for large $n$ is "very close to what would be sufficient". The prime-gap hypothesis is unproved, and the paper proves no unconditional infinitude (Section 4, p. 512).

The paper gives the deduction in one sentence. A check of this page, not of the paper: put $x=2^{k+2}$, let $q$ be the least prime above $x$ and $p$ the largest prime at most $x-b-4$, where $b=q-x$. Under the gap hypothesis, $b<0.1\sqrt x\,(1+o(1))$ and $a=x-p<b+4+0.1\sqrt x\,(1+o(1))$, so $b+3<a<0.21\sqrt x$ for large $k$, while $2^{(k-1)/2}=2^{-3/2}\sqrt x

0.35\sqrt x$; $a$ and $b$ are odd because $p$ and $q$ are odd. So every large $k$ gives one such $n$, and distinct $k$ give distinct $n$ since $2^k$ is the exact power of $2$ dividing $n$.

Almost perfect variant (Section 4, p. 513, a remark without proof). The paper says the proof of Theorem 1 "can be easily adapted" with $2^k$ replaced by an almost perfect number $m$, one with $\sigma(m)=2m-1$: if $p=4m-a$ and $q=4m+b$ are primes for odd positive integers $a,b$ with $b+3<a<\sqrt{m/2}$, then $mpq$ is a primitive weird number. It concludes that an odd almost perfect number larger than $1$, with a corresponding choice of $p$ and $q$, would give an odd weird number. Whether any almost perfect number other than a power of $2$ exists is, the paper notes, unknown. The adapted proof is not written out.

Source. G. Melfi, On the conditional infiniteness of primitive weird numbers, J. Number Theory 147 (2015), 508--514, DOI 10.1016/j.jnt.2014.07.024; Theorem 1 on p. 509, its proof in Section 3 on pp. 510--512, Lemma 2 on p. 510, the conditional consequence on pp. 509--510, the almost perfect remark on p. 513. The edition is recorded on the source card.

Read depth. Claims checked: the statement, the conditional consequence and the almost perfect remark were read clause by clause against the published print; the proof of Theorem 1 was followed for its structure, and the abundance identity below was re-derived, but the interval bounds of the weirdness step were not re-derived. A second reader checked the statement, hypotheses, constants, labels and pages, the conditional consequence, the almost perfect remark and this page's own check of the deduction against the print.

Proof pointer

Section 3, pp. 510--512. The proof assumes $k\ge8$, citing the remark on small $k$ in the introduction, and has three steps.

• Abundant (p. 511): $\sigma(n)=(2^{k+1}-1)(p+1)(q+1)$ gives $\Delta(n)=2^{k+1}(a-b-3)+(a-1)(b+1)$, positive because $a>b+3$.
• Primitive abundant (p. 511): $\Delta(n/p)=2^{k+1}-q-1$ and $\Delta(n/q)=2^{k+1}-p-1$ are negative, and $\Delta(n/2)<0$ uses $a-b-2<2^{(k-1)/2}$ and $(a-1)(b+1)<2^k$. Since every multiple of an abundant number is abundant, $n$ is then a multiple of no smaller weird number, so weirdness makes it primitive weird (p. 510).
• Weird (pp. 511--512): by Lemma 2 (p. 510), an abundant $n$ is weird exactly when $\Delta(n)$ is not a sum of distinct proper divisors of $n$, because $\Delta(n)+n$ is the sum of all proper divisors. The proper divisors of $n$ below $2^{3k/2}$ are the powers $2^j$ with $j\le k$ and the numbers $2^jp$, $2^jq$ with small $j$, so every sum of distinct proper divisors up to $2^{3k/2}$ lies in one of the blocks $I_0=\{1,\dots,2^{k+1}-1\}$ and $I_h=\{hp,\dots,hq+2^{k+1}-1\}$ for $1\le h<2^{(k-1)/2}$. These blocks are pairwise disjoint and increasing, and with $h^*=(a-b)/2-2$ the abundance $\Delta(n)$ lies strictly between $\max I_{h^*}$ and $\min I_{h^*+1}$.

Dependencies

Lemma 2 (p. 510) of the same paper, proved there. For the conditional consequence, the prime-gap hypothesis stated above; Cramér's and Gonek's conjectures and the Baker--Harman--Pintz theorem are the paper's references [5], [7] and [1].

Bears on

• Problem 470: Theorem 1 gives an explicit family of primitive weird numbers, and the conditional consequence answers the second question (infinitely many primitive weird numbers) yes under the unproved bound $p_{n+1}-p_n<0.1\,p_n^{1/2}$ for large $n$; the claim is recorded on the problem's claim page. The first question, on odd weird numbers, is touched only by the almost perfect remark, which turns it into the existence of an odd almost perfect number above $1$ together with suitable primes; it is not answered.